Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations – Exercise 4.1 NCERT Solutions

Introduction

Exercise 4.1 introduces the concept of complex numbers. You will learn how to represent complex numbers, identify real and imaginary parts, and perform basic operations. This exercise builds the foundation for solving quadratic equations that have non‑real roots.

Key Concepts

1. Definition of Complex Number:

$$z=a+ib,a,b\in \mathbb{R},i^2=-1$$

• $a$ = real part ($\mathrm{\Re }(z)$)

• $b$ = imaginary part ($\mathrm{\Im }(z)$)

2. Equality of Complex Numbers:

$$a+ib=c+id⟹a=c,b=d$$

3. Addition/Subtraction:

$$(a+ib)+(c+id)=(a+c)+i(b+d)$$

$$(a+ib)-(c+id)=(a-c)+i(b-d)$$

4. Multiplication:

$$(a+ib)(c+id)=(ac-bd)+i(ad+bc)$$

5. Conjugate:

$$\bar{z}=a-ib$$

NCERT Questions with Solutions (10)

Q1. Express $\sqrt{-4}$ in terms of $i$.

$$\sqrt{-4}=\sqrt{4}\cdot i=2i$$

Q2. Express $\sqrt{-9}$ in terms of $i$.

$$\sqrt{-9}=3i$$

Q3. Express $\sqrt{-25}$ in terms of $i$.

$$\sqrt{-25}=5i$$

Q4. Write real and imaginary parts of $z=3+4i$. $\mathrm{\Re }(z)=3,\mathrm{\Im }(z)=4$

Q5. Write real and imaginary parts of $z=-5+7i$. $\mathrm{\Re }(z)=-5,\mathrm{\Im }(z)=7$

Q6. Write real and imaginary parts of $z=2-9i$. $\mathrm{\Re }(z)=2,\mathrm{\Im }(z)=-9$

Q7. Find conjugate of $z=3+4i$.

$$\bar{z}=3-4i$$

Q8. Find conjugate of $z=-5+7i$.

$$\bar{z}=-5-7i$$

Q9. Add $z_1=3+4i,z_2=1-2i$.

$$z_1+z_2=(3+1)+i(4-2)=4+2i$$

Q10. Multiply $z_1=2+3i,z_2=4-i$.

$$z_1{z}_2=(2\cdot 4-3\cdot (-1))+i(2\cdot (-1)+3\cdot 4)=(8+3)+i(-2+12)=11+10i$$

FAQs (10)

1. What is a complex number? Number of form $a+ib$.

2. What is imaginary unit? $i$ with property $i^2=-1$.

3. What is real part of $z$? Coefficient $a$.

4. What is imaginary part of $z$? Coefficient $b$.

5. What is conjugate of $z$? $\bar{z}=a-ib$.

6. What is equality of complex numbers? Real and imaginary parts must be equal.

7. What is addition of complex numbers? Add real parts and imaginary parts separately.

8. What is multiplication of complex numbers? Use distributive law and $i^2=-1$.

9. What is modulus of complex number? $\mathrm{\mid }z\mathrm{\mid }=\sqrt{a^2+b^2}$.

10. Why is this exercise important? It introduces basics of complex numbers used in quadratic equations.

Conclusion

Exercise 4.1 has 10 solved questions and 10 FAQs that strengthen your understanding of complex numbers and their basic operations. This builds the foundation for quadratic equations with complex roots in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations – Exercise 4.1 NCERT Solutions